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Theorem eldmcoss 38440
Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)
Assertion
Ref Expression
eldmcoss (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Distinct variable groups:   𝑢,𝐴   𝑢,𝑅   𝑢,𝑉

Proof of Theorem eldmcoss
StepHypRef Expression
1 dmcoss3 38435 . . 3 dom ≀ 𝑅 = dom 𝑅
21eleq2i 2831 . 2 (𝐴 ∈ dom ≀ 𝑅𝐴 ∈ dom 𝑅)
3 eldmcnv 38327 . 2 (𝐴𝑉 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
42, 3bitrid 283 1 (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wex 1776  wcel 2106   class class class wbr 5148  ccnv 5688  dom cdm 5689  ccoss 38162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-coss 38393
This theorem is referenced by:  eldmcoss2  38441  eldm1cossres  38442
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